Spectral invariants for monotone Lagrangians

@article{Leclercq2018SpectralIF,
  title={Spectral invariants for monotone Lagrangians},
  author={R'emi Leclercq and Frol Zapolsky},
  journal={Journal of Topology and Analysis},
  year={2018}
}
Since spectral invariants were introduced in cotangent bundles via generating functions by Viterbo in the seminal paper [73], they have been defined in various contexts, mainly via Floer homology theories, and then used in a great variety of applications. In this paper we extend their definition to monotone Lagrangians, which is so far the most general case for which a “classical” Floer theory has been developed. Then, we gather and prove the properties satisfied by these invariants, and which… 

Bounds on spectral norms and barcodes

We investigate the relations between algebraic structures, spectral invariants, and persistence modules, in the context of monotone Lagrangian Floer homology with Hamiltonian term. Firstly, we use

Quantitative Heegaard Floer cohomology and the Calabi invariant

We define a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of

Function theoretical applications of Lagrangian spectral invariants

Entov and Polterovich considered the concept of heaviness and superheaviness by the Oh-Schwarz spectral invariants. The Oh-Schwarz spectral invariants are defined in terms of the Hamiltonian Floer

The spectral diameter of a Liouville domain

The group of compactly supported Hamiltonian diffeomorphisms of a symplectic manifold is endowed with a natural bi-invariant distance, due to Viterbo, Schwarz, Oh, Frauenfelder and Schlenk, coming

Symplectic cohomology and a conjecture of Viterbo

  • E. Shelukhin
  • Mathematics
    Geometric and Functional Analysis
  • 2022
We identify a new class of closed smooth manifolds for which there exists a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in a unit cotangent disk

Invariants of Lagrangian cobordisms via spectral numbers

We extend parts of the Lagrangian spectral invariants package recently developed by Leclercq and Zapolsky to the theory of Lagrangian cobordism developed by Biran and Cornea. This yields a

Lagrangian configurations and Hamiltonian maps

. We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional

Noncontractible Hamiltonian loops in the kernel of Seidel’s representation

The main purpose of this note is to exhibit a Hamiltonian diffeomorphism loop undetected by the Seidel morphism of certain 2-point blow-ups of $S^2 \times S^2$, exactly one of which being monotone.

Bounds on the Lagrangian spectral metric in cotangent bundles

Let $N$ be a closed manifold and $U \subset T^*(N)$ a bounded domain in the cotangent bundle of $N$, containing the zero-section. A conjecture due to Viterbo asserts that the spectral metric for

A Note on Partial Quasi-Morphisms and Products in Lagrangian Floer Homology in Cotangent Bundles

We define partial quasi-morphisms on the group of Hamiltonian diffeomorphisms of the cotangent bundle using the spectral invariants in Lagrangian Floer homology with conormal boundary conditions,

References

SHOWING 1-10 OF 102 REFERENCES

On the Hofer geometry for weakly exact Lagrangian submanifolds

We use spectral invariants in Lagrangian Floer theory in order to show that there exist isometric embeddings of normed linear spaces (finite or infinite-dimensional, depending on the case) into the

New energy-capacity-type inequalities and uniqueness of continuous Hamiltonians

We prove a new variant of the energy-capacity inequality for closed rational symplectic manifolds (as well as certain open manifolds such as cotangent bundle of closed manifolds...) and we derive

Symplectic quasi-states on the quadric surface and Lagrangian submanifolds

The quantum homology of the monotone complex quadric surface splits into the sum of two fields. We outline a proof of the following statement: The unities of these fields give rise to distinct

Rigid subsets of symplectic manifolds

Abstract We show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the

On the extrinsic topology of Lagrangian submanifolds

We investigate the extrinsic topology of Lagrangian submanifolds and of their submanifolds in closed symplectic manifolds using Floer homological methods. The first result asserts that the homology

On exotic monotone Lagrangian tori in CP 2 and S 2 × S 2

Because of Darboux’s theorem and because any open subset of Cn contains a Lagrangian torus, it is possible to construct a Lagrangian torus in any symplectic manifold. The construction of Lagrangian

Spectral Invariants with Bulk, Quasi-Morphisms and Lagrangian Floer Theory

In this paper we first develop various enhancements of the theory of spectral invariants of Hamiltonian Floer homology and of Entovi-Polterovich theory of spectral symplectic quasi-states and

Spectral invariants, analysis of the Floer moduli spaces and geometry of the Hamiltonian diffeomorphism group

In this paper, we apply spectral invariants, constructed in [Oh5,8], to the study of Hamiltonian diffeomorphisms of closed symplectic manifolds $(M,\omega)$. Using spectral invariants, we first

A comparison of symplectic homogenization and Calabi quasi-states

We compare two functionals defined on the space of continuous functions with compact support in an open neighborhood of the zero section of the cotangent bundle of a torus. One comes from Viterbo's

Floer homology and Novikov rings

We prove the Arnold conjecture for compact symplectic manifolds under the assumption that either the first Chern class of the tangent bundle vanishes over π2(M) or the minimal Chern number is at
...